As we have already mentioned, the Stone measure \(\mu \)  defined in the previous paragraph
yields a measure on the \(\sigma \) -algebra formed by Borel sets of Hintikka chains.
This follows from [1]} but we present a short self-contained argument.
Let \({\cal R}\)  be the ring formed by finite unions of basic sets of Hintikka chains;
observe that \({\cal R}\)  consists of finite unions of disjoint basic sets of Hintikka chains.
Let \(X\in {\cal R}\)  be the union of disjoint basic sets corresponding to Hintikka formulas \(\psi _1,\ldots ,\psi _k\)  and
define \(\mu _{{\cal R}}(X)=\mu (\psi _1)+\cdots +\mu (\psi _k)\) .
Clearly, the mapping \(\mu _{{\cal R}}\)  is additive.
Since every countable union of non-empty pairwise disjoint sets from \({\cal R}\)  that is contained in \({\cal R}\)  is finite,
\(\mu _{{\cal R}}\)  is a premeasure.
By Carathéodory's Extension Theorem, the premeasure \(\mu _{{\cal R}}\)  extends to a measure
on the \(\sigma \) -algebra formed by Borel sets of Hintikka chains.
We will also use \(\mu \)  to denote this measure,
which is uniquely determined by the Stone measure \(\mu \) ;
we believe that no confusion can arise by doing us so.
